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Math Help - Difficult integration

  1. #1
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    Difficult integration

    Evaluate 2\pi \int_{\frac{\pi}{8}}^{\frac{3\pi}{8}} \tan x + \sqrt{1 + \sec ^4 x} dx

    i know the final anwser i need to know how to do it. its not for homework or a test i just want to know how to do it
    Last edited by mr fantastic; April 13th 2009 at 05:34 PM. Reason: Trying to bring some order to chaos
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    lets concentrate on \int \tan x \sqrt{1 + \sec^4 x}~dx

    Note that this is the same as \int \frac {\sec^4x \tan x \sqrt{1 + \sec^4 x}}{\sec^4 x}~dx ........i multiplied by \frac {\sec^4 x}{\sec^4 x}

    Now, let u^2 = 1 + \sec^4 x

    \Rightarrow 2u~du = 4 \sec^4 x \tan x~dx

    and \sec^4 x = u^2 - 1

    So our integral becomes:

    \frac 12 \int \frac {u^2}{u^2 - 1}~du

    which is relatively easy to deal with
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  3. #3
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    Wow! . . . great soluton, Jhevon!

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  4. #4
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    indeed haha! actually, this problem came from http://www.mathhelpforum.com/math-he...plication.html

    and look at my long solution, i thought many things at the same time!
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  5. #5
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    Hey Jhevon - Ever been call a Mathimagican
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  6. #6
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Krizalid View Post
    indeed haha! actually, this problem came from http://www.mathhelpforum.com/math-he...plication.html

    and look at my long solution, i thought many things at the same time!
    well, look at that. i never remembered seeing this integral before. how did you?!

    it was probably a mistake that i stumbled on it

    i just wanted to get rid of the square root. so i made what's underneath it u^2. then i realized that i would need a sec^4 x that wasn't there, so i put it there. wonder how it came to me

    Quote Originally Posted by danny arrigo View Post
    Hey Jhevon - Ever been call a Mathimagican
    haha, no, and for good reason
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  7. #7
    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by danny arrigo View Post
    Hey Jhevon - Ever been call a Mathimagican

    I had a student call me that quite often last year.
    I'm not sure what he meant by it.
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  8. #8
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    Quote Originally Posted by matheagle View Post
    I had a student call me that quite often last year.
    I'm not sure what he meant by it.
    Ya me to, especially with nonlinear differential equations. They keep saying that and "Danny magic."
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  9. #9
    MHF Contributor matheagle's Avatar
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    I guess this is magic to them
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  10. #10
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    Quote Originally Posted by matheagle View Post
    I guess this is magic to them
    To paraphrase Arthur C. Clarke: Any sufficiently advanced mathematics is indistinguishable from magic.

    And to paraphrase a Moderator: This thread is gettting way off-topic. The question has been answered so ....... thread closed.
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